49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.

ƒ(x) = 1/3 x³ - 2x² - 5x + 2

검증된 단계별 안내

Start by identifying the function you need to graph: \( f(x) = \frac{1}{3}x^3 - 2x^2 - 5x + 2 \). This is a cubic polynomial function.

Use a graphing utility or software to input the function. This will help you visualize the graph and identify key features such as intercepts, local extrema, and inflection points.

Locate the x-intercepts by setting \( f(x) = 0 \) and solving the equation \( \frac{1}{3}x^3 - 2x^2 - 5x + 2 = 0 \). Use the graphing tool to approximate these values.

Identify the local extrema by finding the derivative \( f'(x) \) and setting it to zero to find critical points. The derivative is \( f'(x) = x^2 - 4x - 5 \). Solve \( x^2 - 4x - 5 = 0 \) to find critical points, and use the second derivative test or the graph to determine if they are maxima or minima.

Determine the inflection points by finding the second derivative \( f''(x) \) and setting it to zero. The second derivative is \( f''(x) = 2x - 4 \). Solve \( 2x - 4 = 0 \) to find potential inflection points, and verify by checking the change in concavity on the graph.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

17m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Graphing Functions

Graphing functions involves plotting the curve of a function on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, where the graph crosses the axes, and understanding the overall shape and direction of the graph. Graphing utilities, like graphing calculators or software, can assist in accurately plotting complex functions.

Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. These points are crucial for understanding the function's behavior and are often used as starting points for sketching graphs.

Local Extreme Values and Inflection Points

Local extreme values are points where a function reaches a local maximum or minimum. These can be found using the first derivative test. Inflection points occur where the function changes concavity, identified using the second derivative. Both concepts are essential for understanding the function's shape and behavior, especially when graphing.

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.ƒ(x) = 1/3 x³ - 2x² - 5x + 2